Prompt Interval Temporal Logic
(extended version)
Dario Della Monica1 , Angelo Montanari2 , Aniello Murano1 , and Pietro Sala3
1
2
Università degli Studi di Napoli “Federico II”, Napoli, Italy,
[email protected],[email protected]
University of Udine, Udine, Italy [email protected]
3
University of Verona, Verona, Italy, [email protected]
Abstract. Interval temporal logics are expressive formalisms for temporal representation and reasoning, which use time intervals as primitive temporal entities.
They have been extensively studied for the past two decades and successfully
applied in AI and computer science. Unfortunately, they lack the ability of expressing promptness conditions, as it happens with the commonly-used temporal
logics, e.g., LTL: whenever we deal with a liveness request, such as “something
good eventually happens”, there is no way to impose a bound on the delay with
which it is fulfilled. In the last years, such an issue has been addressed in automata theory, game theory, and temporal logic. In this paper, we approach it in
the interval temporal logic setting. First, we introduce PROMPT-PNL, a prompt
extension of the well-studied interval temporal logic PNL, and we prove the undecidability of its satisfiability problem; then, we show how to recover decidability
(NEXPTIME-completeness) by imposing a natural syntactic restriction on it.
1
Introduction
Interval temporal logics provide a powerful framework suitable for reasoning about
time. Unlike classic temporal logics, such as Linear Temporal Logic (LTL) [20] and
the like, they use time intervals, instead of time points, as primitive temporal entities.
Such a distinctive feature turns out to be very useful in various Computer Science and
AI application domains, ranging from hardware and real-time system verification to
natural language processing, from constraint satisfaction to planning [1,2,10,19,21,22].
As concrete applications, we mention TERENCE [13], an adaptive learning system for
poor comprehenders and their educators (based on Allen’s interval algebra IA [1]),
and RISMA [16], an algorithm to analyze behavior and performance of real-time data
systems (based on Halpern and Shoham’s modal logic of Allen’s relations HS [14]).
A fundamental class of properties that can be expressed in (both interval- and pointbased) temporal logics is that of liveness properties, which allow one to state that something “good” will eventually happen. However, a limitation that is common to most temporal logics is the lack of support for promptness: it is not possible to bound the delay
with which a liveness request is fulfilled, despite the fact that this is desirable for many
practical applications (see [15] for a convincing argument). To overcome such a shortcoming, a whole body of work has been recently devoted to the study of promptness.
In [4,15], the authors extend LTL with the ability of bounding the delay with which a
2
temporal request is satisfied. In [3], the use of prompt accepting conditions in the context of ω-regular automata is explored by introducing prompt-Büchi automata, whose
accepting condition imposes the existence of a bound on the number of non-accepting
states in between two consecutive occurrences of accepting ones. Prompt extensions of
LTL have also been investigated outside the realm of closed systems. Two-player turnbased games with perfect information have been explored in the prompt LTL setting in
[23]. In [9], the authors lift the prompt semantics to ω-regular games, under the parity
winning condition, by introducing finitary parity games. They make use of the concept
of distance between positions in a play that refers to the number of edges traversed in
the game arena; the classical parity winning condition is then reformulated to take into
consideration only those states occurring with a bounded distance. Such an idea has
been generalised to deal with more involved prompt parity conditions [12,18]. In the
field of formal languages, promptness comes into play in [6], where ωB-regular languages and their automata counterpart, known as ωB-automata, are studied. Intuitively,
ωB-regular languages extend ω-regular ones with the ability of bounding the distance
between occurrences of sub-expressions in consecutive ω-iterations, within each word
of the language. Finally, an extension of alternating-time epistemic temporal logic with
prompt-eventuality has been recently investigated in [5].
In this paper, we show that interval temporal logics can be successfully provided
with a support for prompt-liveness specifications by lifting the work done in [4,15] to
the interval-based setting.
In [4], the language of LTL is enriched with parameterized versions of temporal
modalities F (eventually) and U (until), as well as of the dual modalities G (globally)
and R (release). The resulting logic, called PLTL, features the following parameterized
modalities: F≤x , F>y , G≤y , G>x , U≤x , U>y , R≤y , and R>x , where x ∈ X, y ∈ Y ,
and X and Y are two disjoint sets of bounding variables. Intuitively, a formula F≤x φ
is true if φ is satisfied within x time units, according to the valuation of x (the other parameterized modalities have an analogous interpretation). Thus, PLTL models are LTL
models, i.e., words over the powerset of the set of atomic propositions, enriched with a
valuation for the bounding variables in X ∪ Y . The satisfiability problem for PLTL is
PSPACE-complete, as for LTL. The assumption that X and Y are disjoint is crucial in
retaining decidability. In [15], the authors introduce the logic PROMPT-LTL, which restricts PLTL in three ways: (i) a parameterized version is introduced for the modality F
only (parameterized versions of modalities G, U , and R are not included); (ii) only upper bounds appear in parameterized modalities, i.e., no subscript of the form >x occurs;
(iii) there is only one bounding variable. The restriction imposed by PROMPT-LTL
is less strong than it looks like: as shown in [4], operator F≤x , along with the classic
LTL constructs, is enough to define operators G>x , U≤x , R>x (i.e., all the operators
involving in their subscript variables in X). As PROMPT-LTL enriches LTL with the
ability of limiting the amount of time a fulfillment of an existential request (corresponding to a liveness property) can be delayed, it can be thought of as an extension of LTL
with prompt liveness. In [15], it is shown that reasoning about PROMPT-LTL is not
harder than reasoning about LTL, with respect to a series of basic problems, including
satisfiability (PSPACE-complete).
3
In the present paper, we show how to extend the logic PNL of temporal neighborhood (a well-known fragment of HS whose satisfiability problem is NEXPTIMEcomplete [8]), with the ability of expressing prompt-liveness properties. Following
the approach of [15], we introduce ‘prompt’ versions (i.e., upper bounds only) of all
modalities of PNL. The resulting modality templates are as follows: the prompt-rightadjacency hAx i and the prompt-left-adjacency hAx i, capturing prompt-liveness in the
future and in the past, respectively, as well as the dual modalities [Ax ] and [Ax ]. Intuitively, a modality hAx i (for some upper bound x) forces the existence of an event
starting exactly when the current one terminates and ending within an amount of time
bounded above by the value of x. Similarly, hAx i forces the existence of an event ending exactly when the current one begins and starting at most x time units before the
beginning of the current one. Modalities [Ax ] and [Ax ] express dual properties in the
standard way, namely, [Ax ]ψ stands for ¬hAx i¬ψ and [Ax ]ψ stands for ¬hAx i¬ψ. We
name the proposed logic PROMPT-PNL (Section 2).
We first prove that the future fragment of PROMPT-PNL (PROMPT-RPNL), involving the future modalities hAi, [A], hAx i, and [Ax ] only, is expressive enough to
encode the finite colouring problem, known to be undecidable [17]. Undecidability of
PROMPT-RPNL (and PROMPT-PNL) immediately follows (Section 3). Notably, unlike
LTL, PNL is strictly more expressive than its future fragment RPNL (see [11]); such a
separation result holds between PROMPT-PNL and PROMPT-RPNL as well. Our undecidability result hinges on the unrestricted use of bounding variables within prompt
modalities, which allows one to somehow establish tight bounds for the length of intervals. We show that decidability can be recovered by using two disjoint sets of bounding
variables, one for existential modalities and the other for universal ones. Formulas of
d
the resulting logic, which we name PROMPT-PNL,
enjoy some useful monotonicity
property, i.e., the truth of a formula hAx iψ under a certain interpretation σ(x) of the
bounding variable x implies its truth under every interpretation σ 0 , with σ 0 (x) ≥ σ(x).
d
This allows us to prove a small (pseudo-)model property for PROMPT-PNL,
from which
d
we conclude that the satisfiability problem for PROMPT-PNL is NEXPTIME-complete
(Section 4). Due to lack of space, most of the proofs are in Appendix.
2
The logic PROMPT-PNL
Let us start with some basic notions of interval-based temporal logics. A linear order
D is a pair hD, <i, where D is a set, called domain, whose elements are referred to
as points, and < is a strict total order over D. A (strongly) discrete linear order is a
linear order such that there are only finitely many points in between any two points.
In the rest of the paper, we tacitly assume every domain to be discrete. For the sake of
simplicity, we identify the domain of a linear order with the linear order itself, e.g, we
write “d ∈ D” instead of “d ∈ D”. Let d ∈ D. The successors (resp., predecessors) of
d in D are the points d0 ∈ D such that d < d0 (resp., d0 < d); the immediate successor
(resp., immediate predecessor) of d in D, denoted by succD (d) (resp., predD (d)), is (if
any) the point d0 ∈ D such that d0 is a successor (resp., predecessor) of d in D and no
point d00 ∈ D exists with d < d00 < d0 (resp., d0 < d00 < d). Note that succD (d) (resp.,
predD (d)) is defined unless d is the greatest (resp., least) element in D. Given a linear
4
order D and two points a, b ∈ D, with a < b, we denote by [a, b] an interval (over D).
The set of intervals over a linear order D is denoted by I(D). An interval structure (over
a countable set AP of atomic propositions) is a pair hD, V i, where D is a linear order
and V : I(D) → 2AP is a valuation function, which assigns to each interval over D the
set of atomic proposition that are true over it. Given a linear order D and a, b ∈ D, we
denote by D≥a (resp., D>a , D≤a , D<a , D[a,b] , D]a,b[ , D[a,b[ , D]a,b] ) the set of elements
d ∈ D such that d ≥ a (resp., d > a, d ≤ a, d < a, a ≤ d ≤ b, a < d < b, a ≤ d < b,
a < d ≤ b). For instance, we denote by R>0 the set of positive reals.
Syntax and semantics. Let AP (atomic propositions) and X (bounding variables) be
two countable sets. Formulas of PROMPT-PNL in negation normal form are defined as
follows:
ϕ ::= p | ϕ ∧ ϕ | hAiϕ | hAiϕ | hAx iϕ | hAx iϕ
| ¬p | ϕ ∨ ϕ | [A]ϕ | [A]ϕ | [Ax ]ϕ | [Ax ]ϕ
where p ∈ AP and x ∈ X. We also use other standard Boolean connectives, e.g.,
→, and logical constants > and ⊥, which are defined in the usual way. We denote
by PROMPT-RPNL the PROMPT-PNL fragment obtained by excluding past modalities hAi, [A], hAx i, and [Ax ], and we write PROMPT-(R)PNL when we refer to both
formalisms. In the following, we will take the liberty of writing PROMPT-(R)PNL formulas not in negation normal form when useful.
PROMPT-(R)PNL models are interval structures enriched with a valuation function
for bounding variables in X and a metric over the underlying domain. Formally, a model
for PROMPT-(R)PNL (over AP and X) is a quadruple hD, V, σ, δi, where hD, V i is
an interval structure (D is the domain of the model), σ : X → R>0 is a valuation
function for bounding variables, and δ : D × D → R>0 is a metric over D (i.e., the pair
(D, δ) is a metric space) satisfying the additional properties: for every d, d0 , d00 ∈ D
(i) if d < d0 < d00 , then δ(d, d00 ) = δ(d, d0 ) + δ(d0 , d00 ), (ii) if d has infinitely many
¯ | d < d}
¯ is not bounded above, and (iii) if d has
successors in D, then the set {δ(d, d)
¯ d) | d¯ < d} is not bounded above.
infinitely many predecessors in D, then the set {δ(d,
For a model M = hD, V, σ, δi, we let DM = D, VM = V , σM = σ, and δM = δ,
that is, DM , VM , σM , and δM denote the four components of M . A PROMPT-(R)PNL
model is finite (resp., infinite) if so is its domain.
The truth value of a PROMPT-PNL formula over a model and an interval in it is
inductively defined as follows:
– M, [a, b] |= p if and only if p ∈ VM ([a, b]), for every p ∈ AP;
– M, [a, b] |= ¬p if and only if p 6∈ VM ([a, b]), for every p ∈ AP;
– M, [a, b] |= ϕ1 ∧ ϕ2 if and only if M, [a, b] |= ϕ1 and M, [a, b] |= ϕ2 ;
– M, [a, b] |= ϕ1 ∨ ϕ2 if and only if M, [a, b] |= ϕ1 or M, [a, b] |= ϕ2 ;
– M, [a, b] |= hAiϕ if and only if there is c ∈ D>b
M such that M, [b, c] |= ϕ;
– M, [a, b] |= [A]ϕ if and only if for all c ∈ D>b
M it holds M, [b, c] |= ϕ;
– M, [a, b] |= hAiϕ if and only if there is c ∈ D<a
M such that M, [c, a] |= ϕ;
– M, [a, b] |= [A]ϕ if and only if for all c ∈ D<a
M it holds M, [c, a] |= ϕ;
– M, [a, b] |= hAx iϕ if and only if there is c ∈ D>b
M , with δM (b, c) ≤ σM (x),
such that M, [b, c] |= ϕ, for every x ∈ X;
– M, [a, b] |= [Ax ]ϕ if and only if for all c ∈ D>b
M , with δM (b, c) ≤ σM (x),
it holds M, [b, c] |= ϕ, for every x ∈ X;
5
– M, [a, b] |= hAx iϕ if and only if there is c ∈ D<a
M , with δM (c, a) ≤ σM (x),
such that M, [c, a] |= ϕ, for every x ∈ X;
– M, [a, b] |= [Ax ]ϕ if and only if for all c ∈ D<a
M , with δM (c, a) ≤ σM (x),
it holds M, [c, a] |= ϕ, for every x ∈ X.
The truth value of a PROMPT-RPNL formula is obtained, as expected, by restricting to
the relevant clauses only.
In PNL, modalities hLi and hLi, corresponding to Allen’s relations later and before, are definable as: hLiϕ ≡ hAihAiϕ and hLiϕ ≡ hAihAiϕ. Additionally, in
PROMPT-PNL it is possible to define the ‘prompt’ counterparts of modalities hLi and
hLi as: hLx iϕ ≡ hAx ihAx iϕ and hLx iϕ ≡ hAx ihAx iϕ. The resulting semantic interpretation for hLx i and hLx i is as follows:
– M, [a, b] |= hLx iϕ if and only if there is [c, d] ∈ I(DM ) such that b < c, δM (b, c) ≤
σM (x), δM (c, d) ≤ σM (x), and M, [c, d] |= ϕ;
– M, [a, b] |= hLx iϕ if and only if there is [c, d] ∈ I(DM ) such that d < a,
δM (d, a) ≤ σM (x), δM (c, d) ≤ σM (x), and M, [c, d] |= ϕ.
Intuitively, a modality hLx i, for some bounding variable x, requires the existence of
an event starting and ending within a bounded amount of time after the termination of
the current one (modalities hLx i impose an analogous constraint in the past). Obviously,
only hLx i is definable in PROMPT-RPNL (hLx i is not).
The globally-in-the-future modality [G] is defined as [G]ψ ≡ ψ∧[A]ψ∧[A][A]ψ, for
every PROMPT-PNL formula ψ; analogously the prompt-globally-in-the-future modality [Gx ] is defined as [Gx ]ψ ≡ ψ ∧ [Ax ]ψ ∧ [A][Ax ]ψ, for every PROMPT-PNL formula ψ and x ∈ X. Given a PROMPT-(R)PNL model M , modalities [G] and [Gx ]
[a,b]
[a,b],x
induce the sets GM = {[a, b]} ∪ {[c, d] ∈ I(DM ) | b ≤ c} and GM
= {[a, b]} ∪
{[c, d] ∈ I(DM ) | b ≤ c and δM (c, d) ≤ σM (x)}. We omit the subscript M when it is
clear from the context. For every PROMPT-(R)PNL model M , [a, b] ∈ I(DM ), and
PROMPT-PNL formula ψ, it holds that M, [a, b] |= [G]ψ if and only if M, [c, d] |= ψ
for every [c, d] ∈ G [a,b] and M, [a, b] |= [Gx ]ψ if and only if M, [c, d] |= ψ for every
[c, d] ∈ G [a,b],x . Finally, for a model M and [a, b] ∈ I(DM ), we define the length of
[a, b] (in M ) as the value δM (a, b) and, for every p ∈ AP, if M, [a, b] |= p, then we say
that [a, b] is a p-interval (in M ).
The satisfiability problem. A PROMPT-(R)PNL formula ϕ is satisfiable if, and only
if, there exist a PROMPT-(R)PNL model M and an interval [x, y] in M such that
M, [x, y] |= ϕ. Moreover, a satisfiable formula is said to be finitely satisfiable if there
exists a finite model for it; otherwise it is non-finitely satisfiable. The satisfiability (resp.,
finite satisfiability) problem for PROMPT-(R)PNL consists in deciding whether a given
PROMPT-(R)PNL formula is satisfiable (resp., finitely satisfiable).
3
Undecidability of PROMPT-RPNL
We prove the undecidability of the satisfiability problem for the logic PROMPT-RPNL
(and thus for PROMPT-PNL as well), by a reduction from the finite coloring problem
(FCP) [17]. An instance of FCP (aka finite tiling problem) is a tuple ∆ = hC, H, V, ci ,
cf i, where C is a finite, non-empty set of colours, H, V ⊆ C × C are total binary
relations over the set of colours C, and ci , cf ∈ C are distinguished colours. A solution
6
to ∆ is a pair hC, (K, L)i, where K, L ∈ N and C : {0, . . . , K} × {0, . . . , L} → C is a
colouring function such that C(0, 0) = ci , C(K, L) = cf , and, in addition,
– (C(i, j), C(i + 1, j)) ∈ H, for each i < K and j ≤ L (horizontal constraint), and
– (C(i, j), C(i, j + 1)) ∈ V , for each i ≤ K and j < L (vertical constraint).
FCP consists in establishing whether there are two natural numbers K and L, and a
colouring of the plane {0, . . . , K} × {0, . . . , L} such that horizontal and vertical constraints are fulfilled, and bottom-left and top-right colours are given. CFP is undecidable [17, Proposition 7.2]. We encode CFP by means of a PROMPT-RPNL formula. The
different aspects of the problem are encoded by means of (blocks of) formulas and the
correctness of such partial encodings is testified by the corresponding lemmas below.
Clearly, the conjunction of all these formulas is satisfiable if and only if CFP admits a
solution. In what follows, we fix an interval model M = hD, V, σ, δi.
For every d ∈ D and x ∈ X, we define bσcd (x) = max{δ(d, d0 ) ∈ R>0 | d0 ∈
>d
D and δ(d, d0 ) ≤ σ(x)}. It clearly holds that bσcd (x) ≤ σ(x) and, for every d0 ∈
D≥d , we have that δ(d, d0 ) ≤ σ(x) implies δ(d, d0 ) ≤ bσcd (x). For every x ∈ X, there
is exactly one point d0 ∈ D≥d such that δ(d, d0 ) = bσcd (x); we call such a point the
x-canonical successor of d. The length of an interval [d, d0 ] ∈ I(D), where d0 is the
x-canonical successor of d, is said to be x-canonical, for every x ∈ X.
Let succ-upperbound be the formula [G](hAi> → hAs i>), where s ∈ X.
Lemma 1. If M, [a, b] |= succ-upperbound for some [a, b], then for every c ∈ D≥b
that is not the greatest element in D it holds δ(c, succD (c)) ≤ bσcc (s). Moreover,
let c0 be the x-canonical successor of c. If c0 is not the greatest element in D, then
bσcc (x) + σ(s) > σ(x), for every x ∈ X.
Let less-than(x , y) be the formula [G](hAi> → hAy iauxx ,y ) ∧ [G][Ax ]¬auxx ,y
(it is a parametric formula to be instantiated with some x, y ∈ X).
Lemma 2. If M, [a, b] |= less-than(x , y) for some [a, b], then σ(x) < bσcc (y) holds
for every c ∈ D≥b , unless c is the greatest element in D.
Let ∃-last be the conjunction of the following formulas:
^
¬last ∧ hAihAilast ∧ [G](hAilast →
[A](¬p ∧ [A]¬p))
p∈AP
[G](hAilast → [A]¬hAilast)
[G]((last → hAiunique) ∧ (hAiunique → [A]¬hAiunique))
(1)
(2)
(3)
Lemma 3. If M, [a, b] |= ∃-last for some [a, b], then there is exactly one last-interval
in G [a,b] , say it [c, d]. Moreover, it holds c > b and there is no p-interval starting in c or
after it, for every p ∈ AP \ {last}.
Let a ∈ D and [c, d] ∈ I(D) be the unique last-interval (see Lemma 3). Given
p ∈ AP, a p-chain starting at a (or, simply, p-chain) is a finite sequence of p-intervals
[a0 , b0 ], [a1 , b1 ], . . . , [am , bm ] such that a = a0 , bm = c, and bi = ai+1 for every
i ∈ {0, 1, . . . , m − 1}. Let chain(p, x ) be the parametric formula, to be instantiated
with some p ∈ AP and x ∈ X, defined as the conjunction of the following ones:
succ-upperbound ∧ ∃-last
(4)
7
¬p ∧ hAx ip ∧ [G]((p ∧ ¬hAilast) → hAx ip)
[G](p → p1 ∨ p2 )
[G](hAipi → [Ax ][A]p+
i )
+
−
[Gx ](hAipi → pi )
[G](pi → ¬hAip−
i )
[G](hAip → [Ax ](¬p → [A]¬p))
(5)
(6)
i ∈ {1, 2}
i ∈ {1, 2}
i ∈ {1, 2}
(7)
(8)
(9)
(10)
Lemma 4. If M, [a, b] |= chain(p, x ) for some [a, b], then there is a finite p-chain
starting at b whose intervals have x-canonical length. Moreover, no other p-interval
exists in G [a,b],x besides the ones in such a p-chain.
We now provide an encoding of a finite plane {0, . . . , K} × {0, . . . , L}, for some
K, L ∈ N. The idea is to use a u-chain whose intervals are either tile-intervals, encoding
some point of the finite plane, or ∗-intervals, which are used as separators between rows
of the plane. Let plane be the conjunction of the following formulas:
less-than(s, x ) ∧ less-than(x , y) ∧ chain(u, x ) ∧ chain(row, y)
(11)
hAi∗ ∧ [G]((∗ → hAitile) ∧ (u ∧ hAilast → tile))
(13)
[G](hAi∗ → [Ay ](hAi∗ → row))
(15)
[G]((u ↔ ∗ ∨ tile) ∧ (∗ → ¬tile))
(12)
[G](hAirow → hAi∗)
(14)
Lemma 5. If M, [a, b] |= plane for some [a, b], then there is a finite sequence of points
r
b = p10 < p11 < . . . < p1n1 = p20 < p21 < . . . < p2n2 = p30 < . . . < pr−1
nr−1 = p0 <
. . . < prnr , with r ≥ 1 and ni > 1 for every i ∈ {1, . . . , r} such that: (i) [pi0 , pi1 ] is a
∗-interval and its length is x-canonical, for every i ∈ {1, . . . , r}; (ii) [pij , pij+1 ] is a tileinterval and its length is x-canonical, for every i ∈ {1, . . . , r} and j ∈ {1, . . . , ni −1};
(iii) [pi0 , pi+1
0 ] is a row-interval and its length is y-canonical, for every i ∈ {1, . . . , r −
1}; (iv) M, [prnr , p0 ] is the unique last-interval, for some p0 > prnr . Moreover, no other
∗-interval (resp., tile-interval) exists in G [a,b],x .
The encoding of the finite plane {0, . . . , K}×{0, . . . , L} we have obtained so far is
incomplete, the problem being that rows (row-intervals) do not necessarily contain the
same number of tiles (tile-intervals). In order to overcome such a problem, we introduce
below corr-intervals, which are used to link the ith tile-interval of a row to the ith tileinterval of the next row (if any) and to the ith tile-interval of the previous row (if any).
This will guarantee that each row of our encoding features the same number of tiles.
Let w -def be the conjunction of the following formulas:
less-than(x , w ) ∧ less-than(w , y)
(16)
hAs ihAw i([A](¬last ∧ [A]¬last) ∨ hAi∗-aux)
(18)
[Ay ]¬hAi∗-aux ∧ (hAy i(row ∧ ¬hAx ilast) → hAihAi∗-aux)
(17)
Lemma 6. If M, [a, b] |= plane ∧ w -def for some [a, b], then σ(w) < bσcc (y) ≤
σ(y) < σ(w) + σ(s) for every c ∈ D≥b , unless c is the greatest element in D.
8
Let correspondence be the conjunction of the following formulas:
plane ∧ w -def ∧ less-than(s, z ) ∧ less-than(z , x )
(19)
[G](hAx iu → [Az ]hAz i(u-suffix ∧ (hAx iu ∨ hAilast)))
(20)
[Gs ]¬u-suffix
(21)
[G]((row ∧ ¬hAilast) → corr)
(22)
[G]((hAx itile ∧ hAihAx i∗) → hAy icorr)
(23)
[Gw ]¬corr
(24)
[G](corr → hAitile)
(25)
[G](hAx i(tile ∧ hAx i∗) → [Ay ](corr → hAx i(tile ∧ hAx i∗)))
(26)
[pij , pi+1
j ]
Lemma 7. If M, [a, b] |= correspondence for some [a, b], then
is a corri+1
i
interval, with σ(w) < δ(pj , pj ) ≤ σ(y), for every i ∈ {1, . . . , r − 1} and j ∈
{0, . . . , ni − 1}. Moreover, for every i ∈ {1, . . . , r − 1}, it holds that ni = ni+1 .
Now, let ∆ = hC, H, V, ci , cf i be an instance of FCP and let ϕ∆ be the conjunction
of the following formulas:
correspondence ∧ hAx ici ∧ [Gx ]((tile ∧ hAilast) → cf )
_
^
[Gx ](tile ↔
c) ∧ [G](
¬(c ∧ c0 ))
c∈C
c,c0 ∈C,c6=c0
_
[G](hAx i(tile ∧ hAx itile) →
hAx i(c ∧ hAx ic0 ))
(c,c0 )∈H
_
[Gx ]((hAx itile ∧ hAy icorr) →
(hAx ic ∧ [Ay ](corr → hAx ic0 )))
0
(c,c )∈V
(27)
(28)
(29)
(30)
Lemma 8. The formula ϕ∆ is satisfiable iff the CFP instance ∆ has a positive answer.
Theorem 1. The satisfiability problem for PROMPT-RPNL, and thus the one for
PROMPT-PNL, is undecidable.
4
d
Decidability of PROMPT-PNL
In this section, we show how to restrict the use of prompt modalities to get a fragment
of PROMPT-PNL with a decidable satisfiability problem.
d
We define PROMPT-PNL
as the fragment of PROMPT-PNL obtained by using disjoint sets of bounding variables for existential and universal prompt modalities. Formally, let us partition the set X of bounding variables into sets X3 and X2 . The syntax
d
of PROMPT-PNL
is defined as:
ϕ ::= p | ϕ ∧ ϕ | hAiϕ | hAiϕ | hAx iϕ | hAx iϕ
| ¬p | ϕ ∨ ϕ | [A]ϕ | [A]ϕ | [Ay ]ϕ | [Ay ]ϕ
d
where p ∈ AP, x ∈ X3 , and y ∈ X2 . Since PROMPT-PNL
is a syntactic restriction of PROMPT-PNL, both formalisms share the same semantics. In particular,
d
a PROMPT-PNL model is a PROMPT-PNL
model as well. Analogously to the unred
d
stricted case, we define PROMPT-RPNL as PROMPT-PNL
devoid of past modalities
hAi, [A], hAx i, and [Ay ].
9
d
d
PROMPT-PNL
is not closed under negation. For any given PROMPT-PNL
formula
ψ, we inductively define neg(ψ) as shown in Table 1 (neg(ψ) is not necessarily a
d
PROMPT-PNL
formula). If ψ is a (non-prompt) PNL formula, then neg(ψ) ≡ ¬ψ.
Moreover, we define neg(∼ ψ) as ψ and thus we have that neg(neg(ψ)) ≡ ψ, for every
d
PROMPT-PNL
formula ψ.
ψ
neg(ψ)
ψ
neg(ψ)
p
¬p
¬p
p
ψ1 ∧ ψ2
neg(ψ1 ) ∨ neg(ψ2 )
ψ1 ∨ ψ2
neg(ψ1 ) ∧ neg(ψ2 )
hAiψ1
[A]neg(ψ1 )
[A]ψ1
hAineg(ψ1 )
hAiψ1
[A]neg(ψ1 )
[A]ψ1
hAineg(ψ1 )
ψ
hAx iψ1 or hAx iψ1 or [Ay ]ψ1 or [Ay ]ψ1
neg(ψ)
∼ψ
d
Table 1. Definition of neg(ψ), for a PROMPT-PNL
formula ψ
A close analysis of the proof of the undecidability of PROMPT-(R)PNL reveals that
the unrestricted use of bounding variables within prompt modalities allows one to somehow establish tight bounds for the length of intervals, and this ability is crucial to the
encoding. We are going to show that decidability can be recovered by not allowing both
existential and universal prompt quantification on the same bounding variable. Intuitively, decidability follows from the fact that, when disjoint sets of bounding variables
are used within existential and universal prompt modalities, formulas enjoy a monotonicity property, which does not hold for unrestricted PROMPT-(R)PNL formulas.
Let M = hD, V, σ, δi be a PROMPT-PNL model, x ∈ X, and r ∈ R>0 . We denote
by M[x:=r] the model hD, V, σ 0 , δi, where σ 0 (x) = r and σ 0 (x0 ) = σ(x0 ) for every
x0 ∈ X with x0 6= x.
d
Proposition 1 (monotonicity). Let ψ be a formula of PROMPT-PNL,
M be a model of
d
PROMPT-PNL,
and [a, b] be an interval in M . If M, [a, b] |= ψ, then M[x:=r] , [a, b] |=
ψ for all x ∈ X3 and r ∈ R>0 , with r ≥ σM (x). In a dual fashion, if M, [a, b] |= ψ,
then M[y:=r] , [a, b] |= ψ for all y ∈ X2 and r ∈ R>0 , with r ≤ σM (y).
d
Checking that the above monotonicity property holds for PROMPT-PNL
is immediate.
To see that it does not hold for PROMPT-PNL, consider the formula ψ = [Ay ]¬p ∧
hAx ip ∧ [Ax ]¬q ∧ hAz iq. Clearly, ψ is satisfiable and all of its models are such that the
value of x is bounded below by the value of y and above by the value of z.
d
By Proposition 1, when studying the (finite) satisfiability problem for PROMPT-PNL
we can assume, w.l.o.g., that |X3 | = |X2 | = 1, as every formula ψ, featuring (possibly) more than one bounding variable in X3 or X2 , can be transformed into an equisatisfiable one ψ 0 , obtained by replacing two distinguished (chosen randomly) variables
x̂ ∈ X3 and ŷ ∈ X2 for every x ∈ X3 and y ∈ X2 , respectively. It is not difficult to check that, due to monotonicity, ψ is (finitely) satisfiable if and only if so is ψ 0 .
Therefore, for the remainder of the section, we set X3 = {x} and X2 = {y}.
d
Finite satisfiability. The finite satisfiability problem for PROMPT-PNL
can be reduced
to the one for plain PNL, known to be NEXPTIME-complete [8]. Let ψ be a formula of
d
PROMPT-PNL
and let plain(ψ) be the PNL formula obtained from ψ by:
10
(i) replacing existential prompt modalities by the corresponding non-prompt versions (i.e., substituting hAi for hAx i and hAi for hAx i), and
(ii) replacing all sub-formulas of the forms [Ay ]ψ and [Ay ]ψ by the constant >.
It is not difficult to show by induction on the structure of ψ that if ψ is finitely satisfiable, so is plain(ψ). On the other hand, if plain(ψ) is finitely satisfiable, then let
Mplain(ψ) = hD, V i be a PNL model such that Mplain(ψ) , [a, b] |= plain(ψ) for some
[a, b] ∈ I(D). We define δ(d, d0 ) = |{d00 ∈ D | d < d00 ≤ d0 }| for every d, d0 ∈ D.
Since D is finite, both maxδ = max {δ(d, d0 ) | d, d0 ∈ D and d 6= d0 } and minδ =
min {δ(d, d0 ) | d, d0 ∈ D and d 6= d0 } are well defined, thus we can set σ(x) = maxδ ,
δ
and σ(y) = min
2 . It is possible to show that M = hD, V, σ, δi is such that M, [a, b] |=
ψ. Therefore, ψ is finitely satisfiable, too.
d
Theorem 2. The finite satisfiability problem for PROMPT-PNL
is NEXPTIME-complete.
In order to deal with formulas that are non-finitely satisfiable, in what follows we
show how the search for an infinite model can be reduced to the search for a finite
witness for it, within a finite search space. Decidability of the satisfiability problem for
d
PROMPT-PNL
immediately follows.
4.1
Prompt labeled interval structures
d
In this subsection we define labeled interval structures for PROMPT-PNL
formulas,
which are, intuitively, extended models, where intervals are labeled with sets of subformulas (instead of sets of atomic propositions) of the considered formula. From now
d
on, we let ϕ be a generic PROMPT-PNL
formula.
Let Sub(ϕ) be the set of all sub-formulas of ϕ and let Sub¬ (ϕ) = {neg(ψ) |
ψ ∈ Sub(ϕ)}. The closure of ϕ, denoted by Cl (ϕ), is the set Sub(ϕ) ∪ Sub¬ (ϕ) ∪
{hAiϕ, neg(hAiϕ)}. Clearly, |Cl (ϕ)| ≤ 2 · |ϕ| + 2 holds.
A future temporal request of ϕ is a formula in Cl (ϕ) having one of the following
forms: hAiψ, neg(hAiψ), hAx iψ, neg(hAx iψ), [Ay ]ψ, neg([Ay ]ψ), for some ψ. Analogously, a past temporal request of ϕ is a formula in Cl (ϕ) having one of the following
forms: hAiψ, neg(hAiψ), hAx iψ, neg(hAx iψ), [Ay ]ψ, neg([Ay ]ψ), for some ψ. We
denote by TR f (ϕ) (resp., TR p (ϕ)) the set of future (resp., past) temporal requests
of ϕ. In addition, the set of temporal requests of ϕ, denoted by TR(ϕ), is defined as
TR f (ϕ) ∪ TR p (ϕ).
A ϕ-atom is a subset A of Cl (ϕ) such that, for every ψ, ψ1 , ψ2 ∈ Cl (ϕ), (i) ψ ∈ A
if and only if neg(ψ) ∈
/ A, and (ii) ψ1 ∨ ψ2 ∈ A if and only if ψ1 ∈ A or ψ2 ∈ A.
Notice that conditions (i) and (ii) imply ψ1 ∧ ψ2 ∈ A if and only if ψ1 ∈ A and ψ2 ∈ A.
We denote the set of ϕ-atoms by Aϕ .
A prompt ϕ-labeled interval structure (pLISϕ ) is a 5-tuple L = hD, L, δ, X , Y i,
where (D, δ) is a metric space, L : I(D) → Aϕ is a labeling function (or simply labeling) such that ϕ ∈ L([a, b]) for some [a, b] ∈ I(D), and X , Y ∈ N are the existential
and the universal bound, respectively. Sometimes, for the sake of brevity, we omit the
last three components of the 5-tuple and we denote a pLISϕ as a 2-tuple hD, Li instead.
Moreover, given a pLISϕ L = hD, Li, we denote by DL its underlying domain D and
by LL the labeling function L. A pLISϕ L is finite (resp., infinite) if so is DL .
11
Given a pLISϕ L and a point d ∈ DL we define the set of future requests of
S
d in L, denoted by f-REQL (d), as d0 ∈D<d (LL (d0 , d) ∩ TR f (ϕ)), the set of past
S
requests of d in L, denoted by p-REQL (d), as d0 ∈D>d (LL (d, d0 ) ∩ TR p (ϕ)), and
the set of requests of d in L, denoted by REQL (d), as f-REQL (d) ∪ p-REQL (d).
We denote by REQϕ the class of all sets of requests, i.e., REQϕ = {R | R =
REQL (d) for some pLISϕ L and d ∈ DL }. We have that |REQϕ | ≤ 2|Cl(ϕ)| ≤ 22·|ϕ|+2 .
An existential request of ϕ is a temporal request of ϕ of one the following forms:
hAiψ, hAiψ, hAx iψ, hAx iψ, neg([Ay ])ψ, and neg([Ay ])ψ, for some ψ. A universal request of ϕ is a temporal request of ϕ that is not an existential one. Let L =
hD, L, δ, X , Y i be a pLISϕ and d ∈ D. We define ∃-REQL (d) = {ψ ∈ REQL (d) |
ψ is an existential request of ϕ} and ∀-REQL (d) = REQL (d) \ ∃-REQL (d).
For ψ ∈ ∃-REQL (d), we say that ψ is fulfilled in (L, d) by d0 ∈ D if, and only if,
one of the following holds:
– ψ = hAiψ 0 for some ψ 0 and ψ 0 ∈ L([d, d0 ]),
– ψ = hAiψ 0 for some ψ 0 and ψ 0 ∈ L([d0 , d]),
– ψ = hAx iψ 0 for some ψ 0 , ψ 0 ∈ L([d, d0 ]), and δ(d, d0 ) ≤ X ,
– ψ = hAx iψ 0 for some ψ 0 , ψ 0 ∈ L([d0 , d]), and δ(d0 , d) ≤ X ,
– ψ = neg([Ay ]ψ 0 ) for some ψ 0 , neg(ψ 0 ) ∈ L([d, d0 ]), and δ(d, d0 ) ≤ Y ,
– ψ = neg([Ay ]ψ 0 ) for some ψ 0 , neg(ψ 0 ) ∈ L([d0 , d]), and δ(d0 , d) ≤ Y .
ψ is fulfilled in (L, d) if and only if there is d0 such that ψ is fulfilled in (L, d) by d0 .
For ψ ∈ ∀-REQL (d), we say that ψ is fulfilled in (L, d) if, and only if, one of the
following holds:
– ψ = [A]ψ 0 for some ψ 0 and ψ 0 ∈ L([d, d0 ]) for every d0 ∈ D>d ,
– ψ = [A]ψ 0 for some ψ 0 and ψ 0 ∈ L([d0 , d]) for every d0 ∈ D<d ,
– ψ = [Ay ]ψ 0 for some ψ 0 and ψ 0 ∈ L([d, d0 ]) for every d0 ∈ D>d with δ(d, d0 ) ≤ Y ,
– ψ = [Ay ]ψ 0 for some ψ 0 and ψ 0 ∈ L([d0 , d]) for every d0 ∈ D<d with δ(d0 , d) ≤ Y ,
– ψ = neg(hAx iψ 0 ) for some ψ 0 and neg(ψ 0 ) ∈ L([d, d0 ]) for every d0 ∈ D>d with
δ(d, d0 ) ≤ X ,
– ψ = neg(hAx iψ 0 ) for some ψ 0 and neg(ψ 0 ) ∈ L([d0 , d]) for every d0 ∈ D>d with
δ(d0 , d) ≤ X .
d is ∃-fulfilled in L if, and only if, every ψ ∈ ∃-REQL (d) is fulfilled; d is ∀-fulfilled in
L if, and only if, every ψ ∈ ∀-REQL (d) is fulfilled; d is fulfilled in L if, and only if, it
is both ∃- and ∀-fulfilled in L.
An existentially fulfilling (resp., universally fulfilling, fulfilling) pLISϕ , aka ∃-pLISϕ
(resp., ∀-pLISϕ , ∃∀-pLISϕ ), is a pLISϕ L such that every d ∈ DL is ∃-fulfilled (resp.,
∀-fulfilled, fulfilled) in it.
Proposition 2. ϕ is satisfiable if and only if there exists a ∃∀-pLISϕ , and it is finitely
satisfiable if and only if there exists a finite ∃∀-pLISϕ .
d
Before showing the decidability of PROMPT-PNL,
we prove a result that will later
L
come in handy. A set of requests REQ (d) (for a pLISϕ L and d ∈ DL ) is consistent if
for each ψ ∈ REQL (d), we have that neg(ψ) ∈
/ REQL (d); otherwise, it is inconsistent.
Proposition 3. Let L be a pLISϕ and d ∈ DL . The following properties hold, unless
REQL (d) is inconsistent:
12
– if D<d 6= ∅, then f-REQL (d) = LL (d0 , d) ∩ TR f (ϕ), for any given d0 ∈ D<d ,
unless f-REQL (d) is inconsistent;
– if D>d 6= ∅, then p-REQL (d) = LL (d, d0 ) ∩ TR p (ϕ), for any given d0 ∈ D>d ,
unless p-REQL (d) is inconsistent.
4.2
A bounded witness for non-finitely satisfiable formulas
Let L be a pLISϕ and d ∈ DL . A set of essentials of d (in L) is any minimal (with
respect to set inclusion) set E ⊆ DL such that for every ψ ∈ ∃-REQL (d) there is d0 ∈ E
for which ψ is fulfilled in (L, d) by d0 . We denote by EL (d) the class containing all
sets of essentials of d in L, i.e., EL (d) = {E ⊆ DL | E is a set of essentials of d in L}.
Intuitively, a set of essentials of d is a collection of points that jointly make d ∃-fulfilled
in L. Clearly EL (d) 6= ∅ if and only if d is ∃-fulfilled in L. We lift this concept to a
higher order: a set of essentials of essentials (or 2nd-order essentials) of d (in L) is any
minimal (with respect to set inclusion) set E2 ⊆ DL such that (i) E1 ⊆ E2 for some
E1 ∈ EL (d) and (ii) for every d0 ∈ E1 there is Ed0 ∈ EL (d0 ) for which Ed0 ⊆ E2 .
We denote by EL2 (d) the class containing all sets of 2nd-order essentials of d in L, i.e.,
EL2 (d) = {E ⊆ DL | E is a set of 2nd-order essentials of d in L}.
Definition 1 (representative). Let L be a finite pLISϕ and d ∈ DL .
If d ∈
/ {min DL , max DL }, then a representative of d in L is a point e ∈ DL
such that REQL (d) = REQL (e), e is fulfilled in L, and so are points in E2 , for some
E2 ∈ EL2 (e) with E2 ∩ {min D, max D} = ∅.
If d = min DL (resp., d = max DL ), then a representative of d in L is a point
e ∈ DL that is a representative of d0 in L for some d0 ∈ DL , with p-REQL (d0 ) =
p-REQL (d) (resp., f-REQL (d0 ) = f-REQL (d)).
A convex subset of a domain D is a subset D0 of D such that for every d0 , d00 ∈ D0
and d ∈ D, if d0 < d < d00 , then d ∈ D0 . A right-convex (resp., left-convex) subset of a
domain D is a convex subset D0 of D such that max D ∈ D0 (resp., min D ∈ D0 ).
Given a pLISϕ L and D0 ⊆ DL , we let request-setsL (D0 ) = {R | REQL (d) =
R for some d ∈ D0 }.
Definition 2 (left- and right-periodic pLISϕ ). Let L be a finite pLISϕ . A left-period
for L is a left-convex subset E of DL such that, for every d ∈ E, if d is not fulfilled in L
or d = min E, then there is d0 ∈ E>d for which the following holds:
a) d0 is a representative of d in L;
0
b) request-setsL (E \ {min E}) is equal to request-setsL (E<d \ {min E}), which is
0
L
>d0
00
<d0
equal to request-sets (E ), and there are d ∈ E
\ {min E} and d000 ∈ E>d
L
L 00
L 000
such that p-REQ (min E) = p-REQ (d ) = p-REQ (d );
c) every hAx iψ ∈ f-REQL (d0 ) is fulfilled in (L, d0 ) by a point belonging to E.
A right-period for L is defined symmetrically.
L is periodic if, and only if, there exist both a left- and a right-period for it.
Definition 3 (ϕ-witness). A ϕ-witness is a finite, periodic ∀-pLISϕ L, such that every
d ∈ DL \ (E ∪ F) is fulfilled in L, where E and F are, respectively, a left- and a
right-period for L, with E ∩ F = ∅ and DL \ (E ∪ F) 6= ∅.
13
Lemma 9. An infinite ∃∀-pLISϕ L = hD, L, δ, X , Y i exists if and only if a ϕ-witness
L0 = hD0 , L0 , δ 0 , X 0 , Y 0 i exists.
Thanks to the previous lemma, we can reduce the search for an infinite model for a
formula to the search for a finite witness. However, since such a finite witness can be
arbitrarily large, the search space is still infinite. In what follows, we provide a bound
on the size of the finite witness, thus obtaining a finite search space. Decidability of
d
PROMPT-PNL
immediately follows.
Let Bϕ = |REQϕ | · (2 · |Cl (ϕ)|2 + 2 · |Cl (ϕ)|) + |REQϕ | · |Cl (ϕ)| + |Cl (ϕ)|.
Lemma 10. Let L = hD, L, δ, X , Y i be a ϕ-witness, E and F being, respectively, a
left- and a right-period for it. If |E| > Bϕ (resp., |F| > Bϕ , |D \ (E ∪ F)| > Bϕ ), then
there is a ϕ-witness L0 = hD0 , L0 , δ 0 , X 0 , Y 0 i with |D0 | = |D| − 1.
The size of a pLISϕ L is the size of the underlying domain DL . The following
corollary immediately follows from the above lemma.
Corollary 1 (small model property). A ϕ-witness exists if and only if there is one of
size at most 3 · Bϕ ≤ 3 · [22·|ϕ|+2 · (2 · (2 · |ϕ| + 2)2 + 2 · (2 · |ϕ| + 2)) + 22·|ϕ|+2 · (2 ·
|ϕ| + 2) + (2 · |ϕ| + 2)].
d
Theorem 3. The satisfiability problem for PROMPT-PNL
is NEXPTIME-complete.
5
Conclusions
In this paper, we have studied the problem of enriching the well-known propositional
logic of temporal neighborhood PNL with support for prompt-liveness specifications.
We first proved that the logic obtained from PNL by introducing “prompt” versions
of its modalities with no restriction on the use of bounding variables, that we call
PROMPT-PNL, is undecidable. Then, we showed that decidability can be recovered
by introducing a partition of bounding variables into two classes, one for the existential
modalities, the other for the universal ones. The satisfiability problem for the resulting
d
logic, named PROMPT-PNL,
is indeed NEXPTIME-complete.
The work done can be further developed in various directions.
First, we are interested in identifying the minimum number of bounding variables
that suffice to make PROMPT-PNL undecidable. We believe it possible to prove that
when the set of variables is small enough, e.g., when it includes two bounding variables only, the logic is still expressive enough to capture some meaningful promptness
conditions and remains decidable.
We also aim at investigating the more powerful setting of parametric extensions of
PNL. Parametric PNL can be viewed as a natural generalization of PROMPT-PNL, as
parametric modalities allow one to express both lower and upper bounds on the delay
with which a request is fulfilled (PROMPT-PNL only copes with the latter).
Last but not least, we are interested in comparing the expressiveness of the logics
d
PROMPT-PNL and PROMPT-PNL
with that of metric PNL, that is, the metric extension
of PNL introduced and systematically studied in [7].
14
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16
A
Proofs of Section 3
Lemma 1. If M, [a, b] |= succ-upperbound for some [a, b], then for every c ∈ D≥b
that is not the greatest element in D it holds δ(c, succD (c)) ≤ bσcc (s). Moreover,
let c0 be the x-canonical successor of c. If c0 is not the greatest element in D, then
bσcc (x) + σ(s) > σ(x), for every x ∈ X.
Proof. Let c ∈ D≥b and assume that c is not the greatest element in D. Let succD i (d) be
the result of applying i times succD to d, for every d ∈ D and i ∈ N. Since M, [a, b] |=
succ-upperbound , there is d ∈ D>c , with δ(c, d) ≤ σ(s) (and thus δ(c, d) ≤ bσcc (s)
holds as well), and, by the definition of bσcc (s), it holds δ(c, succD (c)) ≤ δ(c, d) ≤
bσcc (s).
To prove the second part of the claim, observe that, by the definition of bσcc (x),
δ(c, succD (c0 )) > σ(x) holds. Since δ(c, succD (c0 )) = δ(c, c0 ) + δ(c0 , succD (c0 )) and
δ(c0 , succD (c0 )) ≤ bσcc (s) ≤ σ(s), we conclude that bσcc (x) + σ(s) ≥ δ(c, c0 ) +
δ(c0 , succD (c0 )) = δ(c, succD (c0 )) > σ(x).
t
u
Lemma 2. If M, [a, b] |= less-than(x , y) for some [a, b], then σ(x) < bσcc (y) holds
for every c ∈ D≥b , unless c is the greatest element in D.
Proof. Let c ∈ D≥b and let us assume that c is not the greatest element in D. By the
first conjunct of less-than(x , y), there exists d ∈ D>c , with δ(c, d) ≤ σ(y) (and thus
δ(c, d) ≤ bσcc (y) holds as well), such that M, [c, d] |= auxx ,y . By the second conjunct,
if δ(c, d) ≤ σ(x), then M, [c, d] |= ¬auxx ,y . Thus, it must be δ(c, d) > σ(x), and
σ(x) < bσcc (y) follows immediately.
t
u
Lemma 3. If M, [a, b] |= ∃-last for some [a, b], then there is exactly one last-interval
in G [a,b] , say it [c, d]. Moreover, it holds c > b and there is no p-interval starting in c or
after it, for every p ∈ AP \ {last}.
Proof. By the second conjunct of (1) there is a last-interval [c, d], with c ∈ D>b . By the
first conjunct of (1), (2), and (3), there is no other last-interval in G [a,b] . Finally, due to
the third conjunct of (1), for every p ∈ AP there is no p-interval starting in c or after it.
t
u
Lemma 4. If M, [a, b] |= chain(p, x ) for some [a, b], then there is a finite p-chain
starting at b whose intervals have x-canonical length. Moreover, no other p-interval
exists in G [a,b],x besides the ones in such a p-chain.
Proof. By (4) and Lemma 1, bσcc (x) is defined for every c ∈ D≥b and x ∈ X, unless
c is the greatest element in D. By (4) and Lemma 3, there is exactly one last-interval in
G [a,b] , say it [c, d], and we have that c > b. By (5), there is a p-chain [a1 , b1 ], [a2 , b2 ], . . .
starting at b and such that δ(ai , bi ) ≤ bσcai (x) for every i. Since there are only finitely
many points in between b and c, there is an index m for which bm ≥ c. If bm > c, then
M, [am , bm ] |= p ∧ ¬hAilast and, according (5), a p-interval starts at bm , thus after
the beginning of the unique last-interval [c, d]. This is in contradiction with Lemma 3.
Thus, it must be bm = c, and the p-chain is finite, [am , bm ] being its last interval.
17
p
p1
(a)
p
p1
p1
[G](p1 ! ¬hAip1 )
(b)
p
p1
p
p2
[G](p1 ! ¬hAip1 )
p
p1
p1
[G](hAip1 ! [Ax ][A]p+
1)
p+
1
[Gx ](hAip+
1 ! p1 )
(c)
[Gx ](hAip+
1 ! p1 )
p+
1
[G](hAip1 ! [Ax ][A]p+
1)
p
p1
p
p2
p
p2
[G](hAip2 ! [Ax ][A]p+
2)
p+
2
p2
[G](p2 ! ¬hAip2 )
[Gx ](hAip+
2 ! p2 )
Fig. 1. A graphical explanation of how formula chain(p, x ) force a unique ascending chain of
p-labelled intervals. Picture (a) explains how p1 and p2 intervals must be interleaved. Pictures
(b) and (c) show the two cases in which a p-labelled interval placed out of the chain generates a
contradiction.
We show now that, for every i, [ai , bi ] has x-canonical length, that is, δ(ai , bi ) =
bσcai (x). By construction, we already have δ(ai , bi ) ≤ bσcai (x). In order to show that
δ(ai , bi ) ≥ bσcai (x) holds as well we assume, towards a contradiction, that there is a pinterval [a0 , b0 ] ∈ G [a,b],x , such that δ(a0 , b0 ) < bσca0 (x). By (6), [a0 , b0 ] is a p1 - or a p2 interval as well. We assume, without loss of generality, that it is a p1 -interval. Let c0 ∈ D
be such that δ(a0 , c0 ) = bσca0 (x) (the existence of such a point follows from the fact
0
>c0
that bσca0 (x) is well-defined). Due to (7), [c0 , d0 ] is a p+
1 -interval for every d ∈ D
−
0 0
0 0
and, due to (8), [b , c ] is a p1 -interval (notice that it clearly holds δ(b , c ) ≤ σ(x)).
This leads to a contradiction with (9), since the interval [a0 , b0 ] satisfies p1 but it does
not satisfy ¬hAip−
1 . Therefore, we can conclude that the length of every p-interval
[a0 , b0 ] ∈ G [a,b],x is x-canonical, i.e., δ(a0 , b0 ) = bσca0 (x).
To complete the proof, we have to show that there is no other p-interval in G [a,b],x .
Once again, let us assume, towards a contradiction, that there is a p-interval [a0 , b0 ] ∈
G [a,b],x such that [a0 , b0 ] 6= [ai , bi ] for any i ∈ {1, . . . , m}. As we have just shown,
[a0 , b0 ] ∈ G [a,b],x implies δ(a0 , b0 ) = bσca0 (x). Therefore, in order for [a0 , b0 ] to exist, it
must be ai < a0 < bi for some i ∈ {1, . . . , m}. But this contradicts (10), because any
interval ending at ai satisfies hAip but it does not satisfy [Ax ](¬p → [A]¬p) (as [ai , a0 ]
satisfies ¬p and hAip).
As a last note, let us point out that we needed to introduce the auxiliary atomic
propositions p1 and p2 to avoid the following inconsistency. Suppose we replace p
−
(resp., p+ , p− ) for pi (resp., p+
i , pi ) in (7), (8), and (9) and let [ai−1 , bi−1 ] and [ai , bi ] be
two consecutive intervals in the p-chain (thus, bi−1 = ai ). Then, by (7), [succD (ai ), a0 ]
is a p+ -interval for every a0 ∈ D>succD (ai ) , and, by (8), [ai , succD (ai )] is a p− -interval.
Thus, (9) is contradicted, as [ai−1 , bi−1 ] satisfies p but it does not satisfy ¬hAip− .
A graphical account of how formula chain(p, x ) works is given in Figure 1.
t
u
Lemma 5. If M, [a, b] |= plane for some [a, b], then there is a finite sequence of points
r
b = p10 < p11 < . . . < p1n1 = p20 < p21 < . . . < p2n2 = p30 < . . . < pr−1
nr−1 = p0 <
i
r
i
. . . < pnr , with r ≥ 1 and ni > 1 for every i ∈ {1, . . . , r} such that: (i) [p0 , p1 ] is a
18
∗-interval and its length is x-canonical, for every i ∈ {1, . . . , r}; (ii) [pij , pij+1 ] is a tileinterval and its length is x-canonical, for every i ∈ {1, . . . , r} and j ∈ {1, . . . , ni −1};
(iii) [pi0 , pi+1
0 ] is a row-interval and its length is y-canonical, for every i ∈ {1, . . . , r −
1}; (iv) M, [prnr , p0 ] is the unique last-interval, for some p0 > prnr . Moreover, no other
∗-interval (resp., tile-interval) exists in G [a,b],x .
Proof. The last two conjuncts of (11) force the existence of a u- and a row-chain, whose
intervals’ length is x- and y-canonical, respectively (Lemma 4). Moreover, due to the
first two conjuncts of (11), we have that bσcc (s) ≤ σ(s) < bσcc (x) ≤ σ(x) <
bσcc (y) ≤ σ(y) holds for every c ∈ D≥b , unless c is the greatest element in D
(Lemma 2).
By (12), every u-interval is either a ∗- or a tile-interval (it cannot satisfy both ∗
and tile), and, by (13), we have that the first (resp., last) u-interval of the u-chain is a
∗-interval (resp., tile-interval) and that every ∗-interval is followed by a tile-interval.
Thus, there are suitable values for r, n1 , . . . , nr , with r ≥ 1 and ni > 1 for every
i ∈ {1, . . . , r}, such that the u-chain yields a sequence of points p10 < . . . < p1n1 <
. . . < pr0 < . . . < prnr , that satisfies items (i), (ii), and (iv) of the lemma. From (12) it
also follows that no other ∗-interval (resp., tile-interval) exists in G [a,b],x .
We show now that the row-chain can only be arranged in a way that item (iii) is
fulfilled as well. Let us denote the row-chain by [a1 , b1 ], [a2 , b2 ], . . . , [am , bm ]. We want
to show that m = r and [ai , bi ] = [pi0 , pini ] for every i ∈ {1, . . . , r}. From (14), it
clearly follows that every row-interval starts where a ∗-interval starts, i.e., for every
i ∈ {1, . . . , m} there is j ∈ {1, . . . , r} such that ai = pj0 . Thus, it suffices to show that
every row-interval ends where the next ∗-interval starts (m = r immediately follows
from that). Assume, towards a contradiction, that there is i ∈ {1, . . . , m} such that
ai = pj0 (for some j ∈ {1, . . . , r}) but bi 6= pjnj . If bi < pjnj , then a row-interval
starts at bi , but no ∗-interval starts there, yielding a contradiction with (14). If, on the
other hand, bi > pjnj , then by (15) [ai , pjnj ] is a row-interval besides the ones in the
row-chain, yielding a contradiction with Lemma 4. Therefore, item (iii) is satisfied as
well.
t
u
Lemma 6. If M, [a, b] |= plane ∧ w -def for some [a, b], then σ(w) < bσcc (y) ≤
σ(y) < σ(w) + σ(s) for every c ∈ D≥b , unless c is the greatest element in D.
Proof. Let c ∈ D≥b be a point that is not the greatest element in D. The inequalities
σ(w) < bσcc (y) ≤ σ(y) trivially hold by (16) and Lemma 2. In order to prove that
σ(y) < σ(w)+σ(s) holds as well, we proceed as follows. By (17), unless the row-chain
(whose existence is guaranteed by Lemma 5) features only one row-interval (r = 1
in Lemma 5), there is a ∗-aux-interval starting after the end of the first row-interval
(with no ∗-aux-interval starting before, or at the end of, the first row-interval). By (18),
there is an interval [p0 , p00 ], with p0 > p10 , δ(p10 , p0 ) ≤ σ(s), and δ(p0 , p00 ) ≤ σ(w).
Formula (18) also forces [p0 , p00 ] to end after the end of the first row-interval (i.e., p00 >
p1n1 ), by requiring either that p00 is located after the beginning of the unique last-interval
(left-hand side of the disjunction in (18), to deal with the case when the row-chain
features one row-interval only) or that a ∗-aux-interval starts at p00 (right-hand side of
the disjunction in (18), to deal with the case when the row-chain features more than one
19
row-interval). We need to distinguish the two cases because in the former one it is not
possible for p00 to be the starting point of a ∗-aux-interval (as in this case the first rowinterval is immediately followed by the unique last-interval, and thus no ∗-aux-interval
starts after it), while in the latter case it is not possible for p00 to be located after the
beginning of the unique last-interval (as this would make [p0 , p00 ] longer than σ(w)). It
is easy to see that the following holds: δ(p0 , p00 ) = δ(p10 , p1n1 ) − δ(p10 , p0 ) + δ(p1n1 , p00 ).
Since δ(p10 , p1n1 ) + δ(p1n1 , p00 ) > σ(y) and δ(p10 , p0 ) ≤ σ(s), we have that σ(w) ≥
δ(p0 , p00 ) > σ(y) − σ(s), which implies σ(y) < σ(w) + σ(s).
A graphical account of how formula w -def works is given in Figure 2.
t
u
hAs ihAw i([A](¬last ^ [A]¬last) ! hAi ⇤
hAihAi ⇤
aux
b cc (s)
[Ay ]¬hAi ⇤ aux
aux)
⇤
b cc (w)
row
b cc (y)
⇤
aux
row
b cc (x)
Fig. 2. A graphical explanation of how formula w -def force w to be equal to y − 1. In particular,
the case of multiple row-labelled intervals is depicted.
Lemma 7. If M, [a, b] |= correspondence for some [a, b], then [pij , pi+1
j ] is a corri+1
i
interval, with σ(w) < δ(pj , pj ) ≤ σ(y), for every i ∈ {1, . . . , r − 1} and j ∈
{0, . . . , ni − 1}. Moreover, for every i ∈ {1, . . . , r − 1}, it holds that ni = ni+1 .
Proof. First, we observe that for every c ∈ D that is not the greatest element in D
it holds σ(s) < bσcc (z) ≤ σ(z) < bσcc (x) ≤ σ(x) (by (19) and Lemma 2). Formulas (20) and (21) force, for every u-interval, the existence of suffixes that satisfy
i
u-suffix and whose length is suitably bounded. More precisely, let qji ∈ D>pj be
such that δ(pij , qji ) = bσcpij (z), for every i ∈ {1, . . . , r} and j ∈ {0, . . . , ni − 1}.
Then, [qji , pij+1 ] is a u-suffix-interval, and both σ(s) < δ(pij , qji ) ≤ σ(z) and σ(s) <
δ(qji , pij+1 ) ≤ σ(z) hold.
The proof is by induction on j. Due to (19), (22), and Lemma 5, [pi0 , pi+1
0 ] is a
corr-interval for every i ∈ {1, . . . , r − 1}, and thus the lemma is satisfied for j = 0.
In order to proof the inductive case, we proceed as follows. Due to (23), for every
i ∈ {1, . . . , r − 1} and j ∈ {1, . . . , ni − 1} we have that [pij , p0 ] is a corr-interval for
some p0 , with δ(pij , p0 ) ≤ σ(y). Moreover, due to (24), we have that δ(pij , p0 ) > σ(w).
We show that p0 = pi+1
j . Suppose, towards a contradiction, that this is not the case. We
distinguish two cases.
i+1
0
i
0
– If p0 > pi+1
j , then it must be p ≥ pj+1 and the following holds: δ(pj , p ) ≥
i+1
i+1
i+1
i+1
i+1
i+1
i
i
i
i
δ(pj , pj+1 ) = δ(pj−1 , pj−1 ) − δ(pj−1 , pj ) + δ(pj−1 , qj−1 ) + δ(qj−1 , pj ) +
i+1
i+1
i
i
i
δ(pi+1
j , pj+1 ). Since δ(pj−1 , pj−1 ) > σ(w) (by inductive hypothesis), δ(pj−1 , pj )
i+1
i+1
i+1
i+1
i+1 i+1
≤ σ(x), δ(pj−1 , qj−1 ) > σ(s), and δ(qj−1 , pj ) + δ(pj , pj+1 ) > σ(s) +
20
bσcpi+1 (x) > σ(x) (by Lemma 1), we have that δ(pij , p0 ) > σ(w) − σ(x) + σ(s) +
j
σ(x) = σ(w) + σ(s). Moreover, since, by Lemma 6, σ(w) + σ(s) > σ(y), we
have that δ(pij , p0 ) > σ(y), thus getting a contradiction (because δ(pij , p0 ) ≤ σ(y)
holds).
i+1
0
i
0
– If p0 < pi+1
j , then it must be p ≤ pj−1 , and the following holds: δ(pj , p ) ≤
i+1
i+1
i+1
i
i
i
i
i
δ(pj , pj−1 ) = δ(pj−1 , pj−1 ) − δ(pj−1 , pj ). Since δ(pj−1 , pj−1 ) ≤ σ(y) (by inductive hypothesis) and δ(pij−1 , pij ) > σ(s), we have that δ(pij , p0 ) < σ(y) − σ(s).
By Lemma 6, σ(y) − σ(s) < σ(w), thus implying δ(pij , p0 ) < σ(w), yielding a
contradiction (because δ(pij , p0 ) > σ(w) holds).
At last, let i ∈ {1, . . . , r − 1}. We show that ni = ni+1 . Since [pij , pi+1
j ] is a corrinterval for every j ∈ {0, . . . , ni − 1}, it clearly holds ni+1 ≥ ni . Let us assume,
towards a contradiction, that ni+1 > ni . Then, the (i + 1)th row-interval features an exi+1
i
tra tile-interval, namely [pi+1
ni , pni +1 ]. As a consequence, every interval ending in pni −1
leads to a contradiction with (26), because it satisfies hAx i(tile ∧ hAx i∗) but it does not
satisfy [Ay ](corr → hAx i(tile ∧ hAx i∗)) (as the corr-interval [pini −1 , pi+1
ni −1 ] satisfies
i+1 i+1
hAx i(tile ∧ hAx itile), due to the presence of the extra tile-interval, [pni , pni +1 ]). t
u
B
Proofs of Section 4
Proposition 3. Let L be a pLISϕ and d ∈ DL . The following properties hold, unless
REQL (d) is inconsistent:
– if D<d 6= ∅, then f-REQL (d) = LL (d0 , d) ∩ TR f (ϕ), for any given d0 ∈ D<d ,
unless f-REQL (d) is inconsistent;
– if D>d 6= ∅, then p-REQL (d) = LL (d, d0 ) ∩ TR p (ϕ), for any given d0 ∈ D>d ,
unless p-REQL (d) is inconsistent.
Proof. We only prove the first part of the claim (the second one can be proved similarly). By the definition of f-REQL (d), it clearly holds f-REQL (d) ⊇ (LL (d0 , d) ∩
TR f (ϕ)) for any given d0 ∈ D<d . Let us assume that f-REQL (d) is consistent and
let us suppose, towards a contradiction, that f-REQL (d) 6⊆ (LL (d0 , d) ∩ TR f (ϕ)) for
some d0 ∈ D<d . Thus, there is d00 ∈ D<d and a formula ψ such that ψ ∈ LL (d00 , d) ∩
TR f (ϕ) \ LL (d0 , d) ∩ TR f (ϕ), which means ψ ∈ LL (d00 , d) \ LL (d0 , d). By the definition of ϕ-atom, we have that neg(ψ) ∈ LL (d0 , d), as ψ ∈
/ LL (d0 , d) and therefore,
L
by the definition of f-REQ (d), we have that both ψ and neg(ψ) belong to f-REQL (d),
contradicting the consistency hypothesis for f-REQL (d).
t
u
Lemma 9. An infinite ∃∀-pLISϕ L = hD, L, δ, X , Y i exists if and only if a ϕ-witness
L0 = hD0 , L0 , δ 0 , X 0 , Y 0 i exists.
Proof. For a pLISϕ L = hD, L, δ, X , Y i and D̂ ⊆ DL , we denote by L|D̂ the pLISϕ
hD̂, L̂, δ̂, X̂ , Ŷ i, where X̂ = X , Ŷ = Y , and δ̂ and L̂ are the projection of, respectively,
δ and L to the elements of D̂.
(only-if direction) We identify a domain D0 ⊆ D and we show that L|D0 is a ϕwitness. We use the following notations:
21
– B L = {R ∈ REQϕ | ∃d ∈ D.(REQL (d) = R and ∀d0 ∈ D<d .REQL (d) 6= R)},
– B R = {R ∈ REQϕ | ∃d ∈ D.(REQL (d) = R and ∀d0 ∈ D>d .REQL (d) 6= R)}.
– U L = request-setsL (D) \ B L ,
– U R = request-setsL (D) \ B R .
Intuitively, B L (resp., B R ) is the set of left-bounded (resp., right-bounded) requests,
that is, requests that have a leftmost (resp., rightmost) occurrence in L; U L (resp., U R )
is the set of left-unbounded (resp., right-unbounded) requests, that is, requests that do
not have a leftmost (resp., rightmost) occurrence in L.
We obtain D0 through the following steps.
(1) Let dˆmin be the smallest point in D such that REQL (dˆmin ) ∈ B L (if B L = ∅, then
let dˆmin be a randomly-chosen point in D) and dˆmax be the greatest point in D such
that REQL (dˆmax ) ∈ B R (if B R = ∅ or dˆmax < dˆmin , then let dˆmax = dˆmin ).
000
00
0
ˆ
(2) Let d01 , d001 , d000
1 be the greatest points in D such that (i) d1 < d1 < d1 < dmin ,
0
0 ˆ
L
]d000
,d00
[
L
[d00
1
1
1
(ii) δ(d1 , dmin ) > X , and (iii) request-sets (D
) = request-sets (D ,d1 [ )
0 ˆ
= request-setsL (D[d1 ,dmin [ ) = U L .
0
00
000
ˆ
(3) Let d02 , d002 , d000
2 be the smallest points in D such that (i) dmax < d2 < d2 < d2 ,
0
L
]dˆmax ,d02 ]
L
]d02 ,d00
ˆ
2 ])
(ii) δ(dmax , d2 ) > X , and (iii) request-sets (D
) = request-sets (D
000
L
]d00
,d
[
R
= request-sets (D 2 2 ) = U .
S
00 00
(4) For every d ∈ D[d1 ,d2 ] , pick a set Ed ∈ EL2 (d) and let E = d∈D[d001 ,d002 ] Ed . Let
000
000 000
dmin = predD (min(E ∪ {d000
1 , d2 })) and dmax = succD (max(E ∪ {d1 , d2 })).
0
(5) Let D = {d ∈ D | dmin ≤ d ≤ dmax }.
<dˆ
>dˆ
We now prove that L|D0 is a ϕ-witness, E = D0 min and F = D0 max being,
respectively, a left- and a right-period for it. Note that E ∩ F = ∅ and D0 \ (E ∪ F) 6= ∅.
First, we observe that, due to Proposition 3 and to the fact that L is fulfilling, sets of
requests are preserved from L to L|D0 : formally, it holds REQL|D0 (d) = REQL (d)
for every d ∈ D0 \ {dmin , dmax }, as well as p-REQL|D0 (dmin ) = p-REQL (dmin )
and f-REQL|D0 (dmax ) = f-REQL (dmax ). As an immediate consequence, since the labeling L0 of L|D0 preserves the original one L, universal requests are satisfied, that
means that L|D0 is a ∀-pLISϕ . Moreover, due to step (4), D0 contains a set of essen[dˆ
,dˆ
]
tials of d, for every d ∈ D0 \ (E ∪ F) (notice that D0 \ (E ∪ F) = D0 min min ⊆
[d00 ,d00 ]
D0 1 2 ) and thus they are fulfilled in L|D0 . What we are left to do is showing that
E = {d ∈ D0 | dmin ≤ d < dˆmin } and F = {d ∈ D0 | dˆmax < d ≤ dmax } are, respectively, a left- and a right-period for L|D0 . We only show that E is a left-period for L|D0 ;
proving that F is a right-period for L|D0 can be done symmetrically. By the way it is
defined, E is clearly a left-convex subset of D0 . Let us consider a point d ∈ E such
00
that either d is not fulfilled in L|D0 or d = min E. In either case, d ∈ E<d1 : this
is trivially verified if d = min E; otherwise, it follows from the fact that points in
[d00 ,d00 ]
D0 1 2 are fulfilled in L|D0 (due to step (4) and to the fact that L|D0 is a ∀-pLISϕ ).
00 0
It is not difficult to verify that E[d1 ,d1 ] contains a point d0 that is a representative of
00
d in L|D0 . Clearly, d < d0 holds (as d ∈ E<d1 ). Moreover, condition b of Definition 2 is verified thanks to step (2), which forces all sets of requests occurring in E
000 00
0 ˆ
(except for the one associated with min E) to occur both in E]d1 ,d1 [ and in E[d1 ,dmin [ ;
L|D0
as for the set of requests associated with min E, we have that f-REQ
(min E) = ∅,
22
000
00
while p-REQL|D0 (min E) = p-REQL (min E), and thus there exist d00 ∈ E]d1 ,d1 [ and
0 ˆ
d000 ∈ E[d1 ,dmin [ , with p-REQL|D0 (min E) = p-REQL|D0 (d00 ) = p-REQL|D0 (d000 ). Finally, condition c of Definition 2 is verified as well, for every hAx iψ ∈ f-REQL (d0 ):
thanks to step (2), we have δ(d0 , dˆmin ) > X ; moreover, since d0 is fulfilled, there must
0
be a point d00 ∈ D>d such that hAx iψ is fulfilled in (L, d0 ) by d00 , with δ(d0 , d00 ) ≤ X ,
hence it clearly holds d00 < dˆmin , which implies d00 ∈ E.
(if direction) Let E and F be, respectively, a left- and a right-period for L0 and let
b
D = D0 \ (E ∪ F). Intuitively, we first obtain the domain D of L from the domain
D0 of L0 by iterating the two periods E and F, and then we define δ and L as suitable
extensions of, respectively, δ 0 and L0 to the new, extended domain D. Intuitively, L is
defined so that the set of requests of a point in L is the same as the set of requests of
the corresponding point in L0 . This is possible thanks to the next observation, which
follows from Proposition 3:
it is possible, for every d ∈ D, to force f-REQL (d) to be equal to
0
ˆ for some given dˆ ∈ D0 (except for dˆ = min D0 ), by setf-REQL (d),
ˆ for some d00 ; similarly, it is
(†)
ting, for every d0 ∈ D<d , L(d0 , d) = L0 (d00 , d)
L
L0 ˆ
possible to force p-REQ (d) to be equal to p-REQ (d), for some given
dˆ ∈ D0 (except for dˆ = max D0 ).
0
ˆ and p-REQL0 (d),
ˆ
Notice also that we are guaranteed of the consistency of f-REQL (d)
required to apply Proposition 3, because every point in D0 is either fulfilled or a point
with the same set of requests is fulfilled in L0 (by the definition of ϕ-witness).
b × {0}) ∪ (F × N>0 ), the underlying
Formally, we define D as (E × Z<0 ) ∪ (D
ordering relation ≺D being defined as (in the reminder of the proof we use ≺D0 to
denote the ordering relation over D0 and ≺D to denote the one over D, while the symbol
< is used to denote the standard ordering relation over Z): for every (d, k), (d0 , k 0 ) ∈ D
(i) if k < k 0 , then (d, k) ≺D (d0 , k 0 ), and (ii) if k = k 0 , then (d, k) ≺D (d0 , k 0 ) if and
only if d ≺D0 d0 . The metric δ is defined as follows:
– for every (d, k), (d0 , k 0 ) ∈ D, with k = k 0 , we set δ(d0 , d00 ) = δ 0 (d0 , d00 );
b 0)) = δ 0 (max E, min D)
b
– δ((max E, k), (min E, k + 1)) = δ((max E, −1), (min D,
<−1
for every k ∈ Z
;
b 0), (min F, 1)) = δ 0 (max D,
b min F)
– δ((max F, k), (min F, k + 1)) = δ((max D,
>0
for every k ∈ N .
The above definition leaves δ undefined for some (d0 , d00 ) ∈ D × D. However, the value
of δ over those inputs follows from the properties of the metric space we consider; in
particular, it always holds δ(d0 , d00 ) = δ(d00 , d0 ) = δ(d0 , d000 ) + δ(d000 , d00 ) for every d000 ,
with d0 ≤ d000 ≤ d00 .
In what follows, slightly abusing the notation, we sometimes identify (d, k) simply
with d, for (d, k) ∈ D with k ∈ {−1, 0, 1}. Moreover, for every d ∈ D0 that is fulfilled
in L0 , we pick from EL0 (d) a set of essentials of d, and we denote it by Ed . We define
the labeling function L as follows.
b (thus d is fulfilled in L0 , by the definition of ϕ-witness) and every
A. For every d ∈ D
0
d ∈ Ed , we set:
– L([d0 , d]) = L0 ([d0 , d]), if d0 ≺D0 d;
– L([d, d0 ]) = L0 ([d, d0 ]), if d ≺D0 d0 .
23
B. For every d ∈ E, we proceed as follows.
– If d is fulfilled in L0 and d 6= min E, then, for every d0 ∈ Ed and every
k ∈ Z<0 , we set:
• L([(d0 , k), (d, k)]) = L0 ([d0 , d]), if d0 ≺D0 d;
• L([(d, k), (d0 , k)]) = L0 ([d, d0 ]), if d ≺D0 d0 and d0 ∈ E;
b ∪ F.
• L([(d, k), d0 ]) = L0 ([d, d0 ]), if d ≺D0 d0 and d0 ∈ D
– If d is not fulfilled in L0 or d = min E, then let r be the a representative of d in
L0 fulfilling conditions of Definition 2. For every d0 ∈ Er and every k ∈ Z<0 ,
we set:
• L([(d0 , k − 1), (d, k)]) = L0 ([d0 , r]), if d0 ≺D0 r;
• L([(d, k), (d0 , k)]) = L0 ([r, d0 ]), if r ≺D0 d0 and d0 ∈ E;
b ∪ F.
• L([(d, k), d0 ]) = L0 ([r, d0 ]) if r ≺D0 d0 and d0 ∈ D
C. For every d ∈ F, we proceed analogously to item B.
It is important to notice that labels set in the second sub-case of step B do not conflict
with (i.e., do not overwrite) any of the labels set in the previous steps. Clearly there is
not conflict between labels set in the first sub-case of step B and labels set in the second
one. Suppose, towards a contradiction, that there is an interval [(d, k), (d0 , 0)] ∈ I(D)
b and d ∈ Ed0 ) and then overwritten in
whose label is set in step A (because d0 ∈ D
the second sub-case of step B (because d ∈ E is such that either d is not fulfilled in
L0 or d = min E, d0 ∈ Er , and r is a representative of d in L0 fulfilling conditions of
Definition 2). By the properties of r (Definition 1), we know that d0 is fulfilled in L0
and so are points in Ed0 , as well as that min E ∈
/ Ed0 . Since d ∈ Ed0 (by the conditions
for step A), we have that d is fulfilled in L0 and d 6= min E, which contradicts the
conditions for the second sub-case of step B.
The definition of L we provided so far is incomplete, but it is possible to verify
that, by setting X = 2 · |D0 | (and Y = Y 0 ), all existential requests are fulfilled in L,
for every point in D. In order to complete the labeling it is enough, for every interval
[d, d0 ] ∈ I(D) for which L([d, d0 ]) is still undefined, to set L([d, d0 ]) = L0 ([d00 , d000 ]) for
0
0
some d00 , d000 ∈ D0 such that REQL (d) = REQL (d00 ) and REQL (d0 ) = REQL (d000 ).
The resulting pLISϕ L is a ∃-pLISϕ .
It is important to notice that, thanks to Proposition 3 and the above observation (†)
0
following from it, we have that REQL ((d, k)) = REQL (d) holds for every d ∈ E
0
(except for d = min E, for which we have p-REQL ((min E, k)) = p-REQL (min E)).
This is crucial for the fulfillment of existential requests.
Additionally, since L0 is a ∀-pLISϕ (by Definition 3), so is L. The thesis follows.
t
u
Let Bϕ = REQϕ · (2 · |Cl (ϕ)|2 + 2 · |Cl (ϕ)|) + |REQϕ | · |Cl (ϕ)| + |Cl (ϕ)|.
Lemma 10. Let L = hD, L, δ, X , Y i be a ϕ-witness, E and F being, respectively, a
left- and a right-period for it. If |E| > Bϕ (resp., |F| > Bϕ , |D \ (E ∪ F)| > Bϕ ), then
there is a ϕ-witness L0 = hD0 , L0 , δ 0 , X 0 , Y 0 i with |D0 | = |D| − 1.
Proof. We prove the claim under the assumption that |E| > Bϕ . The other cases (|F| >
Bϕ and |D \ (E ∪ F)| > Bϕ ) can be proven analogously and the proof is omitted.
24
For every d ∈ E, we put repr(d) = max({d0 ∈ E | d0 is a representative of d in L})
(repr(d) is the canonical representative of d in L). By Definition 1, for every d, d0 ∈ E
if REQL (d) = REQL (d0 ), then repr(d) = repr(d0 ). Thus REPR = {repr(d) | d ∈ E}
is such that |REPR| ≤ |REQϕ |. Moreover, we fix, for every r ∈ REPR, a set Er2 ∈
2
2
ES
L (r) of 2nd-order essentials of r in L. Clearly, |Er | ≤ |TR(ϕ)| ≤ |Cl (ϕ)| and
2
| r∈REPR Er | ≤ |REQϕ | · |Cl (ϕ)|.
Since |E| > Bϕ there are R ∈ REQϕ and at least (2·|Cl (ϕ)|2 +2·|Cl (ϕ)|+1) points
S
d ∈ E such that REQL (d) = R, d ∈
/ REPR, and d ∈
/ r∈REPR Er2 . We collect such
points in the set REMOVAL and we denote by REMOVAL>d (resp., REMOVAL<d ) the
set {d0 ∈ REMOVAL | d0 > d} (resp., {d0 ∈ REMOVAL | d0 < d}), for every d ∈ D.
Let de ∈ REMOVAL be a point such that there are at least (|Cl (ϕ)|2 + |Cl (ϕ)|)
points in REMOVAL<de and at least (|Cl (ϕ)|2 + |Cl (ϕ)|) points in REMOVAL>de . We
show how to build the desired ϕ-witness L0 , where D0 = D \ {de }, δ 0 is the projection
of δ to D0 , X 0 = |D0 |, and Y 0 = Y . As for L, it is not possible to simply define it as
the projection of L to D0 because the removal of de might generate defects, such as the
presence of a point d ∈ D<de (resp., d ∈ D>de ) for which there is ψ ∈ f-REQL (d)
(resp., ψ ∈ p-REQL (d)) such that de is the only point in D>d (resp., D<d ) that fulfills
ψ in (L, d). In order to prevent such defects, our definition of L0 guarantees that points
that are fulfilled in L are fulfilled in L0 , too. Formally, we define L0 as follow. Consider
a point d ∈ D \ {de }, with d < de (the case with d > de can be dealt with analogously),
that is fulfilled in L, and assume that for some ψ ∈ f-REQL (d) there is no point d0 ∈
D>d \ {de } for which ψ is fulfilled in (L, d) by d0 . We use a point e ∈ REMOVAL>de
to fix such a defect. In order to properly choose such a point e we proceed as follows.
Let Pde = Ede ∩ D<de (past essentials of de in L) and, forSall dp ∈ Pde , let Fdp =
Edp ∩D>dp (future 2nd-order essentials of de in L). Clearly, | dp ∈Pd Fdp | ≤ |Cl (ϕ)|2 .
e
Now, let Fd = Ed ∩ D>d (future essentials of d in L). We have that de ∈ Fd and
|Fd \{de }| < S
|Cl (ϕ)| (as |Fd | ≤ |Cl (ϕ)|). Thus, there exists a point e ∈ REMOVAL>de
such that e ∈
/ dp ∈Pd Fdp and e ∈
/ Fd ; in order to fix the potential defect caused by the
e
removal of de ,we make e play its role: we set L0 ([d, e]) = L([d, de ]) and L0 ([d0 , e]) =
L([d0 , de ]) for every d0 ∈ Pde . Since REQL (de ) = REQL (e) and due to the properties
of e (in particular, it is unessential to the fulfillment of any of the points in Pde ∪{d}), the
new labeling does not introduce any new defect, while preventing the aforementioned
one (ψ is fulfilled in (L0 , d) by e). Other potential defect, involving other requests ψ 0 ∈
REQL (d) or other points f ∈ D, can be dealt with using the same argument, thus
obtaining the pLISϕ L0 enjoying the property: for every d ∈ D \ {de }, if d is fulfilled in
L, the it is fulfilled in L0 , too.
In order to verify that L0 is actually a ϕ- witness, itS
suffices to observe that the new
labeling does not involve any of the points in REPR ∪ r∈REPR Er2 , which means that,
for every d ∈ E \ {de }, if repr(d) ∈ REPR is a representative of d in L, then it is a
representative of d in L0 as well. The thesis follows.
t
u